Question:- If z is complex number such that |z| ≥ 2 . then the minimum value of |z + 1/2|.
(a) is equal to 5/2
(b) lies int interval (1, 2)
(c) is strictly greater than 5/2
(d) is strictly greater than 3/2 but less than 5/2
Solution:- |z| = 2
let z = x + iy
|x + iy| = 2
⇒ x² + y² = 4 . . . (i)
|z| ≥ 2 is the region on or outside circle whose centre is(0, 0) and radius is 2
The minimum value of |z + 1/2| is distance of z, which lies on the circle |z| = 2 from (-1/2, 0).
∴ minimum value of |z + 1/2| = Distance between (-1/2, 0) and (-2, 0)
= √(-2 + 1/2)² + 0 = 3/2
Geometrically:- Min |z + 1/2| = AD
Hence, minimum value of |z + 1/2| lies in the interval (1, 2)